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We have arbitrary binomials $X_n$ with $EX_n\to \infty$ and $\frac{Var(X_n)}{EX_n^2}\to 0$ . Show that

$$\frac{X_n-EX_n}{\sqrt{Var(X_n)}}\stackrel{d}{\to}N(0,1).$$

Attempt

1) Let $X_n=\sum_{k=1}^n Y_{n,k}$, where iid $Y_{n,k}\sim Bern(p_n)$ then

$Ee^{i\frac{t}{\sqrt{n}}\frac{Y_{n,k}-p_n}{\sqrt{p_n q_n}}}=1+0-\frac{t^2}{2n}\cdot 1+o(\frac{t^2}{n^2}),$

since $E[(\frac{Y_{n,k}-p_n}{\sqrt{p_n q_n}})^2]=1$. So by independence

$$Ee^{it \frac{X_n-EX_n}{\sqrt{Var(X_n)}}}=(1-\frac{t^2}{2n}\cdot 1+o(\frac{t^2}{n^2}))^n\to e^{-t^2/2},$$

since $o\left(\frac{t^2}{n^2}\right)n\to 0$.

But I didn't use any of the two limits i.e.

$np_n\to \infty$ and $\frac{1}{1+\frac{np_n}{q_n}}\to 0\Leftrightarrow \frac{np_n}{q_n}\to \infty$. So there is probably an error.

2) The $X_n=\sum_{k=1}^n Y_{n,k}$, where iid $Y_{n,k}\sim Bern(p_n)$ is set up for Lindeberg-Feller theorem since for each n the $Y_{n,k}$ are independent. We have $$\sum_{k=1}^n E \left(\frac{Y_{n,k}-p_n}{\sqrt{np_n q_n}}\right)^2=1$$ and

$$\sum_{k=1}^n E(\frac{Y_{n,k}-p_n}{\sqrt{np_n q_n}})^21_{|\frac{Y_{n,k}-p_n}{\sqrt{np_n q_n}}|\geq \varepsilon}$$

$$=\frac{1}{np_n q_n} \left[np_n^2 P\left(1\geq \sqrt{\frac{nq_n}{p_n}}\varepsilon,Y_n=0\right)+nq_n^2 P\left(1\geq \sqrt{\frac{np_n}{q_n}}\varepsilon,Y_n=1\right)\right]$$

$$=p_nP(1\geq \sqrt{\frac{nq_n}{p_n}}\varepsilon)+q_n P(1\geq \sqrt{\frac{np_n}{q_n}}\varepsilon).$$

Here the problematic term is $p_n P(1\geq \sqrt{\frac{nq_n}{p_n}}\varepsilon)$ because if $q_{n}\to 0$ like $\frac{1}{n^2}$ then the limits are satisfied but $$p_n P(1\geq \sqrt{\frac{nq_n}{p_n}}\varepsilon)\to 1.$$

Can you provide some hints and corrections? Please not the solution. Thanks

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